How does the spatial process change with:
source: airnow.gov
To rewrite the constrained optimization in terms of \(l_i\) we get \[-\sum_{i=0}^n \sum_{j=0}^n a_i a_j \gamma(\boldsymbol{s_i} - \boldsymbol{s_j}) = -\sum_{i=1}^n \sum_{j=1}^n l_i l_j \gamma_{ij} + 2 \sum_{i=1}^n l_i \gamma_{0i},\] where \(\gamma_{ij} = \gamma(\boldsymbol{s}_i - \boldsymbol{s}_j)\) and hence \(\gamma_{0j} = \gamma(\boldsymbol{s}_0 - \boldsymbol{s}_j)\)
Consider a small example on 1-dimension.
What should the predictions be at \(\boldsymbol{s}^{*}_1 = 4\) and \(\boldsymbol{s}^{*}_2 = 6\)
Define \(\gamma(h) = 1 - \exp(- \frac{h}{3})\) and compute the BLUPs for \(\boldsymbol{s}_1^{*}\) and \(\boldsymbol{s}_2^{*}\)
Interpret and explain \(\boldsymbol{l}\) for each sample point.
If you have time, fill in the line (rather than the surface) from (0.5, 7.5)
As \(\{ E[(\boldsymbol{Y}(\boldsymbol{s_0})|y] - h(\boldsymbol{y}) \}^2 \geq 0\)
Hence to minimize \(E[(\boldsymbol{Y}(\boldsymbol{s_0}) - h(\boldsymbol{y}))^2 | \boldsymbol{y}]\), we set …
\(h(\boldsymbol{y}) = E[(\boldsymbol{Y}(\boldsymbol{s_0})|\boldsymbol{y}]\)
Hence, \(h(\boldsymbol{y})\) that minimizes the error is the conditional expectation of \(\boldsymbol{Y}(\boldsymbol{s_0})\)
Note this is also the posterior mean of \(\boldsymbol{Y}(\boldsymbol{s_0})\)
The conditional distribution, \(p(\boldsymbol{Y_1}| \boldsymbol{Y_2})\) is normal with:
\(E[\boldsymbol{Y_1}| \boldsymbol{Y_2}] = \boldsymbol{\mu_1} + \Omega_{12} \Omega_{22}^{-1} (\boldsymbol{Y_2} - \mu_2)\)
\(Var[\boldsymbol{Y_1}| \boldsymbol{Y_2}] = \Omega_{11} - \Omega_{12} \Omega_[22]^{-1} \Omega_{21}\)
Thus with \(\boldsymbol{Y_1} = Y(\boldsymbol{s_0})\) and \(\boldsymbol{Y_2} = \boldsymbol{y}\) \ \[\Omega_{11} = \sigma^2 + \tau^2, \; \; \; \Omega_{12} = (\sigma^2 p(\phi;d_{01})), \dots, p(\phi;d_{0n}))), \;\; \Sigma_{22} = \sigma^2 H(\phi) + \tau^2 I\]
The conditional distributions are very similar to what we have derived above, watch for HW question.
This can be done with least-squares methods or in a Bayesian framework.